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Paul Hamacher

Publications and source records attributed to Paul Hamacher.

13 recordsLinked to original sources
arXiv2025

Point Counting on Igusa Varieties for function fields

Igusa varieties over the special fibre of Shimura varieties have demonstrated many applications to the Langlands program via Mantovan's formula and Shin's point counting method. In this paper we study Igusa varieties over the moduli stack of global $\Gscr$-shtukas and (under certain conditions) calculate the Hecke action on its cohomology. As part of their construction we prove novel results about local $G$-shtukas in both equal and unequal characteristic and also discuss application of these results to Barsotti-Tate groups and Shimura varieties.

math.AG↗
arXiv2021

On $\mathsf{G}$-isoshtukas over function fields

In this paper we classify isogeny classes of global $\mathsf{G}$-shtukas over a smooth projective curve $C/\mathbb{F}_q$ (or equivalently $σ$-conjugacy classes in $\mathsf{G}(\mathsf{F} \otimes_{\mathbb{F}_q} \overline{\mathbb{F}_q})$ where $\mathsf{F}$ is the field of rational functions of $C$) by two invariants $\barκ,\barν$ extending previous works of Kottwitz. This result can be applied to study points of moduli spaces of $\mathsf{G}$-shtukas and thus is helpful to calculate their cohomology.

math.AG↗
arXiv2020

Finiteness properties of affine Deligne-Lusztig varieties

Affine Deligne-Lusztig varieties are closely related to the special fibre of Newton strata in the reduction of Shimura varieties or of moduli spaces of $G$-shtukas. In almost all cases, they are not quasi-compact. In this note we prove basic finiteness properties of affine Deligne-Lusztig varieties under minimal assumptions on the associated group. We show that affine Deligne-Lusztig varieties are locally of finite type, and prove a global finiteness result related to the natural group action. Similar results have previously been known for special situations.

math.AG↗
arXiv2019

Irreducible components of minuscule affine Deligne-Lusztig varieties

We examine the set of $J_b(F)$-orbits in the set of irreducible components of affine Deligne-Lusztig varieties for a hyperspecial subgroup and minuscule coweight $μ$. Our description implies in particular that its number of elements is bounded by the dimension of a suitable weight space in the Weyl module associated with $μ$ of the dual group.

math.AG↗
arXiv2019

$l$-adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients

Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum of Hodge type. Let $p$ be an odd prime such that $\mathsf{G}_{\mathbb{Q}_p}$ splits after a tamely ramified extension and $p\nmid |π_1(\mathsf{G}^{\rm der})|$. Under some mild additional assumptions that are satisfied if the associated Shimura variety is proper and $\mathsf{G}_{\mathbb{Q}_p}$ is either unramified or residually split, we prove the generalisation of Mantovan's formula for the $l$-adic cohomology of the associated Shimura variety. On the way we derive some new results about the geometry of the Newton stratification of the reduction modulo $p$ of the Kisin-Pappas integral model.

math.NT↗
arXiv2018

On the Newton stratification in the good reduction of Shimura varieties

I construct a generalisation of Mantovan's almost product structure to Shimura varieties of Hodge type with hyperspecial level structure at $p$ and deduce that the perfection of the Newton strata are pro-étale locally isomorphic to the perfection of the product of a central leaf and a Rapoport-Zink space. The almost product formula can be extended to obtain an analogue of Caraiani's and Scholze's generalisation of the almost product structure for Shimura varieties of Hodge type.

math.AG↗
arXiv2017

The almost product structure of Newton strata in the Deformation space of a Barsotti-Tate group with crystalline Tate tensors

In this paper, we construct the almost product structure of the minimal Newton stratum in deformation spaces of Barsotti-Tate groups with crystalline Tate tensors, similar to Oort's and Mantovan's construction for Shimura varieties of PEL-type. It allows us to describe the geometry of the Newton stratum in terms of the geometry of two simpler objects, the central leaf and the isogeny leaf. This yields the dimension and the closure relations of the Newton strata in the deformation space. In particular, their nonemptiness shows that a generalisation of Grothendieck's conjecture of deformations of Barsotti-Tate groups with given Newton polygon holds. As an application, we determine analogous geometric properties of the Newton stratification of Shimura varieties of Hodge type and prove the equidimensionality of Rapoport-Zink spaces of Hodge type.

math.AG↗
arXiv2014

The geometry of Newton strata in the reduction modulo $p$ of Shimura varieties of PEL type

In this paper we study the Newton stratification on the reduction of Shimura varieties of PEL type with hyperspecial level structure and the Newton stratification on the deformation space of a Barsotti-Tate group with PEL structure. Our main result is a formula for the dimension of Newton strata and the description their closure in each of the two cases. Furthermore, we calculate the dimension of some Rapoport-Zink spaces as an intermediate result.

math.AG↗
arXiv2012

The p-rank stratification on the Siegel moduli space with Iwahori level structure

Our concern in this paper is to describe the p-rank statification on the Siegel moduli space with Iwahori level structure over fields of positive characteristic. We calculate the dimension of the strata and describe the closure of a given stratum in terms of p-rank strata. We also examine the relationship between the p-rank stratification and the Kottwitz-Rapoport stratification.

math.AG↗